A classification of subgroups of the Monster isomorphic to S4 and an application

Abstract

We classify all subgroups of the Monster isomorphic to S 4 . We then use this classification to prove that there are no maximal subgroups of the Monster with socles isomorphic to PSU 3 (3), PSL 3 (3), PSL 2 (17), or PSL 2 (7).

Explicit representations of maximal subgroups of the Monster

“…Inside A 12 we already see an involution centralising the first A 6 and swapping the other two in such a way that the 5-point A 5 s get swapped also. We have seen in our investigation of (A 7 × (A 5 × A 5 ):2 2 ):2 that there is an involution centralising the A 7 (and therefore centralising one of the other A 6 factors in our A 6 3 ), swapping the A 5 factors. But in our notation these two A 5 s are both 5-point A 5 s in their respective A 6 s. Hence the normaliser of A 6 3 contains A 6 S 3 .…”

“…We have seen in our investigation of (A 7 × (A 5 × A 5 ):2 2 ):2 that there is an involution centralising the A 7 (and therefore centralising one of the other A 6 factors in our A 6 3 ), swapping the A 5 factors. But in our notation these two A 5 s are both 5-point A 5 s in their respective A 6 s. Hence the normaliser of A 6 3 contains A 6 S 3 . Moreover, the subgroup (S 5 × S 6 × S 6 ) 1 2 inside (S 5 × S 12 ) 1 2 extends this to a welldefined group A 6 3 :S 4 , namely the unique subgroup of index 2 in S 6 S 3 which contains A 6 S 3 .…”

“…By the work of several authors [6,[8][9][10][13][14][15]18] over many years, the classification of the maximal subgroups of the Monster into conjugacy classes is almost complete. There are 43 conjugacy classes of known maximal subgroups, as shown in Table 1, and any other maximal subgroup is almost simple, with socle isomorphic to one of the following simple groups: L 2 (13), U 3 (4), U 3 (8), Sz(8).…”

“…The trickiest case to describe is the normaliser of A 6 3 . The actual structure is as follows: note that S 6 S 3 has a normal subgroup A 6 3 with quotient 2 S 3 ∼ = 2 × S 4 , so there are two subgroups of shape A 6 3 :S 4 . We take the one which contains A 6 S 3 , and adjoin an automorphism which extends all three A 6 factors to M 10 .…”

Maximal subgroups of sporadic groups

“…This was then extended to a systematic study of subgroups generated by two copies of A 5 with 5B-elements [21], which turned up new maximal subgroups L 2 (71) and L 2 (19):2. Later, Holmes [17] classified subgroups isomorphic to S 4 , and used this to classify subgroups isomorphic to…”